Question

if z=1-i then principal value of argz=? (a)3π/4 (b)-3π/4 (c)-π/4 (d)none of them​

Answer 1

Answer:

-pi/4

Step-by-step explanation:

I hope it helps...

Answer 2

$\large\underline{\sf{Solution-}}$

Given complex number is

$\rm :\longmapsto\:z = 1 - i$

To find the argument of z, let convert z in to polar form.

Let assume that

$\rm :\longmapsto\: 1 - i = r(cos \theta + i \: sin \theta )$

$\rm :\longmapsto\: 1 - i = rcos \theta + i \: r \: sin \theta$

On Comparing, real part and Imaginary part, we have

$\purple{\rm :\longmapsto\:rcos \theta = 1 - - - (1)}$

$\purple{\rm :\longmapsto\:rsin \theta = - 1 - - - (2)}$

Squaring equation (1) and (2) and add, we get

$\rm :\longmapsto\: {r}^{2} {cos}^{2} \theta + {r}^{2} {sin}^{2} \theta = 1 + 1$

$\rm :\longmapsto\: {r}^{2}( {cos}^{2} \theta + {sin}^{2} \theta ) = 2$

$\rm :\longmapsto\: {r}^{2} = 2$

$\bf\implies \:r = \sqrt{2}$

On substituting the value of r in equation (1) and (2), we get

$\rm :\longmapsto\:cos \theta = \dfrac{1}{ \sqrt{2} } \: \: and \: \: sin \theta = - \dfrac{1}{ \sqrt{2} }$

$\rm\implies \: \theta = - \: \dfrac{\pi}{4}$

$\bf\implies \: arg(z) = - \: \dfrac{\pi}{4}$

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SHORT CUT TRICK TO FIND ARGUMENT

$\begin{gathered}\boxed{\begin{array}{c|c} \bf Complex \: number & \bf arg(z) \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf x + iy & \sf  {tan}^{ - 1}\bigg |\dfrac{y}{x} \bigg|   \\ \\ \sf  - x + iy & \sf \pi - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf  - x - iy & \sf  - \pi + {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \\ \\ \sf x - iy & \sf  - {tan}^{ - 1}\bigg |\dfrac{y}{x}\bigg | \end{array}} \\ \end{gathered}$